Two Categorical Variables Sections 25.1, 25.2, 25.3, 25.4...

Two Categorical VariablesSections 25.1, 25.2, 25.3, 25.4, 25.8

Lecture 45

Robb T. Koether

Hampden-Sydney College

Mon, Apr 11, 2016

Robb T. Koether (Hampden-Sydney College)Two Categorical VariablesSections 25.1, 25.2, 25.3, 25.4, 25.8Mon, Apr 11, 2016 1 / 28

Outline

1 Two Categorical Variables

2 Expected Counts

3 The χ2 Statistic

4 The χ2 Test

5 Example

6 Assignment


Outline


2 Expected Counts

3 The χ2 Statistic

4 The χ2 Test

5 Example

6 Assignment


Two Categorical Variables

Given two categorical variables, such as sex and politicalaffiliation, we may wonder whether they are related.If they are not related, then we say that they are independent.If sex and political affiliation are independent, then we should seethe same split between Republican, Democrat, and Independentamong men as we see among women.That is, the proportions should be equal.Likewise, we should see the same male/female split whether weare looking at Republicans, Democrats, or Independents.


Two-Way Tables

Suppose we survey 1000 individuals and note their sex and theparty affiliation (Rep, Dem, Ind).We may display the results in a two-way table.

Rep Dem Ind

Total

Male 108 92 200

400

Female 112 218 270

600Total 220 310 470 1000

We have the row totals.We have the column totals.We have the grand total.


Two-Way Tables


Rep Dem Ind TotalMale 108 92 200 400Female 112 218 270 600

Total 220 310 470 1000

We have the row totals.

We have the column totals.We have the grand total.


Two-Way Tables


Rep Dem Ind TotalMale 108 92 200 400Female 112 218 270 600Total 220 310 470

1000

We have the row totals.We have the column totals.

We have the grand total.


Two-Way Tables


Rep Dem Ind TotalMale 108 92 200 400Female 112 218 270 600Total 220 310 470 1000

We have the row totals.We have the column totals.We have the grand total.


Outline


2 Expected Counts

3 The χ2 Statistic

4 The χ2 Test

5 Example

6 Assignment


The Expected Counts

We need to compare the observed counts to the expected counts.Consider the first column, the Republicans.

There were 220 Republicans.Overall, the sample was 40% male and 60% female.Assuming independence, we would expect 40% of the Republicansto be male and 60% to be female.


The Expected Counts

So the expected count of Republican males is

E = 40% of 220

=

(400

1000

)× 220

=400× 220

1000

=Row total× Column total

Grand total

.


The Expected Counts

So the expected count of Republican males is

E = 40% of 220

=

(400

1000

)× 220

=400× 220

1000

=Row total× Column total

Grand total.


Two-Way Tables


Rep Dem Ind TotalMale 108 92 200 400

88 124 188

Female 112 218 270 600

132 186 282

Total 220 310 470 1000


Two-Way Tables



88

124 188

Female 112 218 270 600

132 186 282

Total 220 310 470 1000


Two-Way Tables



88

124 188

Female 112 218 270 600132

186 282

Total 220 310 470 1000


Two-Way Tables



88 124

188

Female 112 218 270 600132

186 282

Total 220 310 470 1000


Two-Way Tables



88 124

188

Female 112 218 270 600132 186

282

Total 220 310 470 1000


Two-Way Tables



88 124 188Female 112 218 270 600

132 186

282

Total 220 310 470 1000


Two-Way Tables



88 124 188Female 112 218 270 600

132 186 282Total 220 310 470 1000


Outline


2 Expected Counts

3 The χ2 Statistic

4 The χ2 Test

5 Example

6 Assignment


The Chi-Square Statistic

To measure how close the observed counts (O) are to theexpected counts (E), we compute the fraction

(O − E)2

E

for each cell in the table.The chi-square statistic χ2 is the sum of these fractions:

χ2 =∑

all cells

(O − E)2

E.


The Chi-Square Distribution

The distribution of the χ2 statistic is not symmetric.Rather, it is skewed right.It also has a different shape for each table size.Thus, we must specify the number of degrees of freedom.



0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1 degree of freedom



1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

2 degree of freedom



2 4 6 8

0.05

0.10

0.15

0.20

3 degree of freedom



2 4 6 8 10 12

0.05

0.10

0.15

4 degree of freedom



2 4 6 8 10 12 14

0.05

0.10

0.15

5 degree of freedom



5 10 15

0.02

0.04

0.06

0.08

0.10

0.12

6 degree of freedom



5 10 15 20

0.02

0.04

0.06

0.08

0.10

0.12

7 degree of freedom



5 10 15 20

0.02

0.04

0.06

0.08

0.10

8 degree of freedom



5 10 15 20 25

0.02

0.04

0.06

0.08

0.10

9 degree of freedom



5 10 15 20 25 30

0.02

0.04

0.06

0.08

10 degree of freedom



5 10 15 20 25 30

0.02

0.04

0.06

0.08

10 degree of freedom vs. N(10,√

20)



10 20 30 40 50 60

0.01

0.02

0.03

0.04

0.05

0.06


40)



20 40 60 80

0.01

0.02

0.03

0.04

0.05


60)



Characteristics of the χ2 distributions:The mean of χ2 equals the degrees of freedom df .The standard deviation of χ2 equals

√2df .

The shape is skewed right, but as df increases, the shapeapproaches the normal distribution N(df ,

√2df ).


Degrees of Freedom

How many degrees of freedom are there in a two-way table? Andwhy are they called “degrees of freedom?”Suppose we know the row and column totals, but not the counts.

Rep Dem Ind TotalMale 400Female 600Total 220 310 470 1000

How many count values can we fill in before the remaining countsare “forced?”


Degrees of Freedom

In a two-way table, the number of degrees of freedom is

df = (No. of rows− 1)× (No. of columns− 1).


Computing the χ2 Statistic

Example (Computing χ2)

Calculate χ2 for the following table.Rep Dem Ind Total

Male 108 92 200 40088 124 188

Female 112 218 270 600132 186 282

Total 220 310 470 1000




χ2 =∑

all cells

(O − E)2

E

=(108− 88)2

88+

(92− 124)2

124+

(200− 188)2

188

+(112− 132)2

132+

(218− 186)2

186+

(270− 282)2

282= 4.545 + 8.258 + 0.766 + 3.030 + 5.505 + 0.511= 22.615.




χ2 =∑

all cells

(O − E)2

E

=(108− 88)2

88+

(92− 124)2

124+

(200− 188)2

188

+(112− 132)2

132+

(218− 186)2

186+

(270− 282)2

282

= 4.545 + 8.258 + 0.766 + 3.030 + 5.505 + 0.511= 22.615.




χ2 =∑

all cells

(O − E)2

E

=(108− 88)2

88+

(92− 124)2

124+

(200− 188)2

188

+(112− 132)2

132+

(218− 186)2

186+

(270− 282)2

282= 4.545 + 8.258 + 0.766 + 3.030 + 5.505 + 0.511

= 22.615.




χ2 =∑

all cells

(O − E)2

E

=(108− 88)2

88+

(92− 124)2

124+

(200− 188)2

188

+(112− 132)2

132+

(218− 186)2

186+

(270− 282)2

282= 4.545 + 8.258 + 0.766 + 3.030 + 5.505 + 0.511= 22.615.


Outline


2 Expected Counts

3 The χ2 Statistic

4 The χ2 Test

5 Example

6 Assignment


The chi2 Test

Our procedure will follow the same 6 steps as always.1. State the hypotheses.2. Give the value of α.3. Write the formula for the test statistic.4. Calculate the value of the test statistic.5. Calculate the p-value.6. Draw a conclusion.


The chi2 Test

The null hypothesis says that there is no difference in thedistributions among the rows or among the columns.That is, the two variables are independent.

H0 : The variables are independent

The alternative hypothesis says the opposite.

Ha : The variables are not independent


The chi2 Test

The test statistic is

χ2 =∑

all cells

(O − E)2

E.

The degrees of freedom is

df = (No. of rows− 1)× (No. of columns− 1).

To find the p-value of χ2, use the χ2cdf function on the TI-83.


Outline


2 Expected Counts

3 The χ2 Statistic

4 The χ2 Test

5 Example

6 Assignment



Example (Computing χ2)Test whether a person’s sex and a person’s political affiliation areindependent.


88 124 188Female 112 218 270 600

132 186 282Total 220 310 470 1000



Example (Computing χ2)(1)

H0 : The variables are independentHa : The variables are not independent

(2) Let α = 0.05.



Example (Computing χ2)(1) The test statistic is

χ2 =∑

all cells

(O − E)2

E.

(2) We calculate χ2 = 22.615.(3) The p-value is

p-value = χ2cdf(22.615,E99,2)

= 1.228× 10−5.

(4) Reject H0 and conclude that sex and political affiliation are notindependent.


Outline


2 Expected Counts

3 The χ2 Statistic

4 The χ2 Test

5 Example

6 Assignment


Assignment

AssignmentRead Sections 25.1, 25.2, 25.3, 25.4, 25.8.Apply Your Knowledge: 1, 2, 3, 5, 6.Check Your Skills: 19, 20, 21, 24, 25.Exercises 30, 31, 32, 34, 35.


Two Categorical Variables Sections 25.1, 25.2, 25.3, 25.4...

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